DATA2002 • Semester 2 • Manav Khemchandani

The Price of a Dream

Exploring what drives the cost of a dream house in the city of dreams.

Introduction

🏠 Property Size & Land Characteristics

Focus: Do bigger houses and lots actually mean higher prices?

Lot.Size — land parcel size
Land.Value — value of land alone
Living.Area — square footage of living space
Bedrooms — number of bedrooms

✨ Quality, Features & Amenities

Focus: Does having more amenities or newer construction raise house prices?

Bathrooms — number of bathrooms
Rooms — total number of rooms
Fireplaces
New.Construct — 1 = new construction, 0 = not new

🌇 Location & Environmental Factors

Focus: Do neighbourhood and environmental factors matter?

Waterfront — on waterfront or not
Pct.College — % adults with college education (proxy for affluence)
Sewer.Type — Public, Private, or None/Unknown

🔧 Age & Structural Characteristics

Focus: Do older homes lose value, and does heating/cooling system influence price?

Age — age of the house
Central.Air — Yes/No
Fuel.Type — Gas, Oil, Electric
Heat.Type — Hot Air, Hot Water, Electric, etc.

1 Acreage vs. Apartments Showdown: Lots, Land, and the Cost Battle

1.1 Lot Size & Living Area

:::::

1.2 Bedrooms & Land Value

:::

1.3 In conclusion,

Living Area and Bedrooms are moderately correlated (≈ 0.66), suggesting overlap.
Land Value and Living Area are related (≈ 0.42) but both add explanatory power.
Overall, Living Area and Land Value dominate housing price variation, while Lot Size contributes little once these are accounted for.

2.1 Amenities, Features & Qualities

The Perks Premium: Do Amenities Add Value?

Box plot on amenities

Linear Regression results:

$Price = 42113.84 + 89371.81Bathroom$
$Price = 171820.23 + 66743.97Fireplace$

2.2 Rooms and New Construction

Box plot on Number of Rooms and Newly Constructed or not

Linear Regression results:

$Price = 52624.99 + 22616.82Room$
$Price = 208580.78 + 73726.04NewConstruct$

2.3 Multivariable regression

Multivariable linear regression: How does the price differ when all variables are combined?

$Predicted Price = -240.6 + 59749.4Bathrooms + 19561.7Fireplaces + 12322.5Rooms + 663.1 NewlyConstruct$

Log-linear regression: Are there any outliers?

$Multivariable R^2 = 0.4341173$

$LogLinear R^2 = 0.4192378$

3.1 Environmental & Neighbourhood Factors

Beyond the Walls: Location as Value

Key variables

Waterfront (binary: Yes/No)
Pct.College (% college-educated residents)
Sewer.Type (Public / Private / None-Unknown)

3.2 Price Relationships

Waterfront vs Price

Neighborhood Education Quality vs Price

3.3 In conclusion,

T-tests show a significant waterfront premium
Regression confirms Pct.College is a strong positive predictor of price
ANOVA shows Public sewer systems linked to higher prices
Combined regression: all three remain independent and significant
Bottom Summary: Neighbourhood quality and infrastructure are systematically capitalised into housing prices.

Combined Effects on Housing Price

4.1 Age & Structural Characteristics vs. Price

Aging Gracefully? The Price of Time

4.2 Fuel Type & Heat Type

4.3 In conclusion,

5.1 Cross Group Analysis

Figure 1: Correlation Matrix of each pair of variables

5.2 Investigation Assumptions

Figure 2: Assumptions plots before and after log transformation

5.3 Findings

Log-Adjusted Model: R-squared: ~ $0.49$ , Adjusted R-Squared: ~ $0.49$

$\text{Predicted Price}=11.2+4.01\times 10^{-04}\cdot \text{Living.Area}+0.14\cdot \text{Bathrooms}$

Simple Comparison Model: R-squared: ~ $0.47$ , Adjusted R-Squared: ~ $0.47$

$\text{Predicted Price}=11.28+5.06\times 10^{-04}\cdot \text{Living.Area}$

Interaction Model: R-squared: ~ $0.49$ , Adjusted R-Squared: ~ $0.49$

$\text{Predicted Price}=11.08+4.77\times 10^{-04}\cdot \text{Living.Area}+0.2\cdot \text{Bathrooms}-3.40\times 10^{-05}\cdot \text{Living.Area:Bathrooms}$

Figure 3: Interactions Summary

Log-Adjusted VIF:

Living.Area   Bathrooms 
   2.067436    2.067436

Examining the p-values in the multiple regression, we see observe a relatively larger p-value for Bathrooms compared to Living.Area although both are significantly low in absolute terms. Comparing with a simple regression on just Living.Area we find that the $r^2$ values are very similar absolutely (diff~0.02) and in proximity with their adjusted $r^2$ ’s, which suggests that although Bathrooms does play a part in the model, its effect is far weaker than that of Living.Area.

However the interaction model paints a slightly different picture, with the plot tapering at the top. The p-value of the interaction between Living.Area and Bathrooms at 0.03 suggests that it is a significant interaction term although small, with the interaction summary likewise finding the p-value for Bathrooms being magnitudes larger than Living.Area’s.

Peeking at the VIF of our model, we see that there it does not suggest any significant multicollinearity issues, corroborating with our findings.

Conclusion

$R^2=0.5905676$

Top 5 Biggest Factors

Living Area / 1000 $ft^2$
Land Value / 10000$
Bathrooms
Waterfront
Lot Size

$Predicted Price = 1346.1 + 69205.5 Living.Area + 9133.5 Land Value + 27682.6 Bathrooms + 123373.7 Waterfront + 7369.6 Lot.Size$

$MultiVarR^2=0.6354036$

$LogLinearR^2=0.5627405$

DATA2002 • Semester 2 • GROUP L21G03

The City of Dreams

Our analysis confirms: The only way to afford a 2-bedroom apartment in Manhattan is if you are a fictional character from a 90s sitcom or the result of a statistically improbable outlier.